User blog:FarEastwood/Probability densities for Mozu's default class growths
Preface A couple weeks ago, a friend and I challenged each other to complete a hard, classic Conquest run with roster restrictions. I challenged him to beat the game using exclusively non-noble characters (other than Corrin), and he challenged me to beat the game using only ranged weapons. I mentioned in passing that our runs shouldn't be too difficult since we both have access to Mozu, and then he expressed frank disappointment with Mozu's offensive viability in his initial play-through. He also commented that she looks more boring than the NPCs, which is hilariously true. For those who aren't aware, Mozu is one of those underdog characters who have the potential to blossom into greater units. Somewhat analogous characters from previous titles include Nino and Est. Now, I'm a big fan of underdogs with obscure potentials—even if they never end up with a true blossoming—so I was admittedly bothered when my friend candidly dismissed Mozu as a poor unit. Later, I found that a noticeable proportion of FE players share the same opinion as my friend—that Mozu kinda sucks. Was I just lucky with my previous Mozu's then? I've been under the assumption that Mozu is a necessity to cleanly complete those harder Conquest maps (e.g., Chapter 17, Chapter 19, Chapter 25), but maybe I'd just been ignorantly overusing one of my favorite characters rather than considering other strategies. With this in mind, I plotted probability densities for Mozu's stat growths for her base and alternate class sets. I wanted to examine how probable it is for Mozu (with Aptitude) to get X-amount of level-up points in each stat for 19 level-ups in each of her default classes. I wanted to know how statistically capable she is as an offensive unit. In a strategy game where tactical advantage and victory pivots about a chance-based system generating binary yes-hits and no-misses, a unit is only as good as the cards she is dealt. And the worst part is: the cards rarely come from a fair deck. Introduction Probabilities in level-ups Any chance-based event in Fire Emblem games essentially utilizes a random number generator (RNG) whose yield results in one of two mutually exclusive outcomes (i.e., a positive or a non-positive) at variable frequencies depending on the probabilities of those outcomes. : Let's say we have an RNG that spits out a random integer between 1 and 10 (inclusive). We need an integer of 4 or smaller to hit; any other integer does not result in a hit. There are 10 possible outcomes from our RNG, and 4 of those outcomes are successful hits. Here, we can say that we have a 40% chance of hitting and 60% chance of not hitting. Of course, most Fire Emblem games utilize RNGs in more convoluted ways than simple random integer generators, but that's a story for a different place and time. Gaining stat points upon a unit's level-up is a series of chance-based events in Fire Emblem games. Each of the unit's stats has a probability to increase and a complementary probability to not increase. The probabilities of increase are listed in a unit's stat growths. Let's examine Mozu's default classes' growth rates. We'll consider stat growth bonuses Mozu's near-exclusive, legitimate access to Aptitude to define our examination. Growth rates: We'll assume that each point-gain opportunity is an independent experiment. The probability of gaining a strength point will not influence the probability of gaining a magic point; not gaining any strength points for several level-ups will not affect the chances of gaining strength points in subsequent level-ups. With all of this in mind, we can see that Mozu has noticeably positive growth rates in her default class sets. ≥70% chance to increase skill and ≥75% chance to increase speed at every level-up in every one of her default classes? Although the numbers look decent, we must remember that they're just probabilities; there is a chance, albeit minimal, for Archer Mozu to get zero speed growth at every level-up. Moreover, stat increases are additive, and considering Mozu’s low base stats, she may need an appreciable number of favorable level-ups to impress the crowds: So how can we know for certain that Mozu will get good stats in the end? Well, we can't—not for certain anyway. We can, however, forecast Mozu's probable finalized stats using simple statistics. Probability densities Given a definite number of attempts at any type of chance-based event, we can readily predict its most probable end-game outcome so long as the details of the chance-based event are simple enough. : Consider an experiment with three flips of a loaded coin. Each coin has a 70%, or 0.7, chance of landing heads (H) and a 30%, or 0.3, chance of landing tails (T). So what are the chances at getting exactly zero H's? One? Two? All three? : When we're only considering the end results of an experiment, we can draw up a forecast chart like the one above. We can interpret the numbers to draw up conclusions like: "the probability of getting at least two H's from this experiment is 78.4%." Gaining stat increases per level-up in Fire Emblem is much like getting multiple heads in the example above. Each character stat has a binary set of outcomes during level-ups where the odds may be skewed towards one of the outcomes. As such, we can draw up forecasts for finalized character stats—granted, we would have to get more creative in coming up with an effective way to (1) get the numbers and (2) present the data since the experiment is much longer than the previous example. I've hopefully addressed both issues in the boring methods section outlined below. Methods Probability density function Probability density functions for Mozu's finalized stats after 19 level-ups were determined using the formula below. Stat increase probabilities (p) were taken from various articles on this wikia. The formula only applies to units that level up 19 times in a single class. Probability = 19Ck × (p)k × (1-p)19-k C = combination formula k = # level-up points p = stat increase probability Data analysis Probability density functions were plotted using google spreadsheets. The distributions were not purely and exclusively normal, so quantile values were examined. 10th and 90th percentile values were chosen to highlight the middle 80th percentile of each density function; while not exactly representative of the most likely outcome of each distribution, the middle 80th-percentile sets an above average distribution range about the median value. Non-integer quantile values were rounded down. Additionally: * modal values along with their probabilities were noted. * probability of getting at least 10 points in each stat were noted. Results | width="50%" valign="top" | |} | width="50%" valign="top" | |} | width="50%" valign="top" | |} | width="50%" valign="top" | |} | width="50%" valign="top" | |} | width="50%" valign="top" | |} Data Talk For those unfamiliar with percentiles, a percentile value essentially represents a proportion of observations in a given dataset. : Say, you and a bunch of students take a 10-point test, and half of all of the scores were at or below 1 point. If you had scored a 2 or above, then you scored better than the 50th percentile score (i.e., half of all the test-takers). If you had scored a one or zero, then you scored within the 50th percentile score range. In the data tables above, we noted the 10th and 90th percentile values so that we can examine the range in between the two: the middle 80th percentile range. The values in this range represent the values that occur in 80% of every possible level 1 through 20 series. We can say that they're 80% likely to occur at the conclusion of Mozu's standard level-ups, with even lesser values occurring 10% of the time and even greater values occurring the final 10% of the time. The data can talk for themselves, so I'll say little. We can see that all of Mozu's middle 80th percentile values hover more or less about (albeit, usually above) the 10-point value. So what are the chances for a stat to gain at least 10 points at the conclusion of its level-ups? Other than her defensive stats in HP, def, and res, Mozu's stats have about an 80% chance to increase by at least 10 points in every possible level-up series. Based on all of this, I would propose using Mozu's str, skl, and spd growths as reference models for any future character examinations. Conclusion So does this mean that people need to be lucky in order to get a Mozu with good stats? Probably not; they don't need much luck when the chances of getting many positives are heavily skewed in their favor. She may need lucky rolls in order to get great defensive stats, but she'll often see noticeably positive growths in her offensive stats even without the need for prayer. So does this mean that you should use Mozu? Well, yes and no. Fire Emblem Fates is a heavily single-player, strategy game that is generous enough to offer many, many roster options for you, the player, to use. Use your favorite units—even if they might not be the best at their role. Use the characters that you like—whether it's based on their design, their stats, or their personality. While I could go on and on about how so-and-so is amazing at such-and-such role, it means very little if you've also beaten the game using what's-her-face doing the exact same thing. Should you use Mozu? Yes, if you want to; no, if you don't. Thanks for taking the time to read this post. Category:Blog posts